chain map

Overview

A chain map is a morphism in the category of chain complexes. It is a map that preserves the differential between the abelian groups.

Definition

A chain map is a function between chain complexes such that

1. is a group homomorphism.
2. is “-linear”. That means the following diagram commutes.

that is,

Connection to homology

A chain map induces a homomorphisms of graded abelian groups

such that there is an induced map

in particular

In short, a chain map induces a morphism on homology groups for a chain complex.

Proof

Let

Pick a representative with . Then

That means . Note that if , then

Thus,

It follows that we can define

Quasi-isomorphism

A morphism of complexes is called a quasi-isomorphism if each morphism

is an isomorphism.

Note: it is called a quasi-isomorphism since it may not be an isomorphism on the level of the complexes. It only becomes an isomorphism when you descend to homology and lose some of the information coming from the complexes.

Example

• Any isomorphism of complexes is a quasi-isormorphism

• Consider the morphism of complexes of abelian groups

passing this to homology we get

and the vertical maps are isomorphisms, but on the chain map above, they are not.